The Magnus expansion, trees and Knuth's rotation correspondence

TL;DR

This paper explores the structure of the Magnus expansion using combinatorics on trees, providing new formulas and insights into its algebraic properties and connections to the Baker-Campbell-Hausdorff series.

Contribution

It introduces a closed formula for the Magnus expansion via planar rooted trees and links it to dendriform algebra structures and the BCH series.

Findings

Derived a closed formula for the Magnus expansion using tree combinatorics

Connected the Magnus expansion to dendriform algebra structures

Revealed new algebraic insights into the BCH series

Abstract

W. Magnus introduced a particular differential equation characterizing the logarithm of the solution of linear initial value problems for linear operators. The recursive solution of this differential equation leads to a peculiar Lie series, which is known as Magnus expansion, and involves Bernoulli numbers, iterated Lie brackets and integrals. This paper aims at obtaining further insights into the fine structure of the Magnus expansion. By using basic combinatorics on planar rooted trees we prove a closed formula for the Magnus expansion in the context of free dendriform algebra. From this, by using a well-known dendriform algebra structure on the vector space generated by the disjoint union of the symmetric groups, we derive the Mielnik-Pleba\'nski-Strichartz formula for the continuous Baker-Campbell-Hausdorff series.